Ratbay Myrzakulov

Modified Gravity in Space-Time with Curvature and
Torsion: F(R, T) Gravity
Ratbay Myrzakulov ∗
Eurasian International Center for Theoretical Physics and Department of General
& Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan
Abstract
In this report, we consider a theory of gravity with a metric-dependent torsion namely
the F(R, T) gravity, where R is the curvature scalar and T is the torsion scalar. We study
the geometric root of such theory. In particular we give the derivation of the model from the
geometrical point of view. Then we present the more general form of F(R, T) gravity with
two arbitrary functions and give some of its particular cases. In particular, the usual F(R)
and F(T) gravity theories are particular cases of the F(R, T) gravity. In the cosmological
context, we find that our new gravitational theory can describe the accelerated expansion of
the Universe.
Contents
1 Introduction 1
2 Preliminaries of F(R), F(G) and F(T) gravities 3
2.1 F(R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 F(T) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 F(G) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Geometrical roots of F(R,T) gravity 4
3.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 FRW case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Lagrangian formulation of F(R,T) gravity 7
5 Cosmological solutions 11
5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6 Conclusion 12
1 Introduction
In the last years the interest in modified gravity theories like F(R) and F(G)-gravity as alternatives
to the ΛCDM Model grew up. Recently, a new modified gravity theory, namely the F(T)-theory,
has been proposed. This is a generalized version of the teleparallel gravity originally proposed
by Einstein [13]-[24]. It also may describe the current cosmic acceleration without invoking dark
energy. Unlike the framework of GR, where the Levi-Civita connection is used, in teleparallel
gravity (TG) the used connection is the Weitzenböck’one. In principle, modification of gravity
may contain a huge list of invariants and there is not any reason to restrict the gravitational
∗ Email: rmyrzakulov@gmail.com; rmyrzakulov@csufresno.edu
1

theory to GR, TG, F(R) gravity and/or F(T) gravity. Indeed, several generalizations of these
theories have been proposed (see e.g. the quite recent review [9]). In this paper, we study some
other generalizations of F(R) and F(T) gravity theories. At the beginning, we briefly review the
formalism of F(R) gravity and F(T) gravity in Friedmann-Robertson-Walker (FRW) universe.
The flat FRW space-time is described by the metric
ds 2 = −dt 2 + a 2 (t)(dx 2 + dy 2 + dz 2 ), (1.1)
where a = a(t) is the scale factor. The orthonormal tetrad components e i (x µ ) are related to the
metric through
g µν = η ij e i µe j ν , (1.2)
where the Latin indices i, j run over 0...3 for the tangent space of the manifold, while the Greek
letters µ, ν are the coordinate indices on the manifold, also running over 0...3.
F(R) and F(T) modified theories of gravity have been extensively explored and the possibility
to construct viable models in their frameworks has been carefully analyzed in several papers (see
[9] for a recent review). For such theories, the physical motivations are principally related to the
possibility to reach a more realistic representation of the gravitational fields near curvature singularities
and to create some first order approximation for the quantum theory of gravitational fields.
Recently, it has been registred a renaissance of F(R) and F(T) gravity theories in the attempt to
explain the late-time accelerated expansion of the Universe [9].
PROBLEM:
Construct such F(R,T) gravity which contents F(R) gravity and F(T) gravity as particular
cases.
In the modern cosmology, in order to construct (generalized) gravity theories, three quantities
– the curvature scalar, the Gauss –Bonnet scalar and the torsion scalar – are usually used (about
our notations see below):
R s = g µν R µν , (1.3)
G s = R 2 − 4R µν R µν + R µνρσ R µνρσ , (1.4)
T s = S ρ µν T ρ µν. (1.5)
In this paper, our aim is to replace these quantities with the other three variables in the form
R = u + g µν R µν , (1.6)
G = w + R 2 − 4R µν R µν + R µνρσ R µνρσ , (1.7)
T = v + S ρ µν T ρ µν, (1.8)
where u = u(x i ;g ij , g˙
ij , g¨
ij ,...;f j ), v = v(x i ;g ij , g˙
ij , g¨
ij ,...;g j ) and w = w(x i ;g ij , g˙
ij , g¨
ij ,...;h j ) are
some functions to be defined. As a result, we obtain some generalizations of the known modified
gravity theories. With the FRW metric ansatz the three variables (1.3)-(1.5) become
R s = 6(Ḣ + 2H2 ), (1.9)
G s = 24H 2 (Ḣ + H2 ), (1.10)
T s = −6H 2 , (1.11)
where H = (lna) t . In the contrast, in this paper we will use the following three variables
R = u + 6(Ḣ + 2H2 ), (1.12)
G = w + 24H 2 (Ḣ + H2 ), (1.13)
T = v − 6H 2 . (1.14)
Finally we would like to note that we expect w = w(u,v). This paper is organized as follows. In
Sec. 2, we briefly review the formalism of F(R) and F(T)-gravity for FRW metric. In particular,
2

the corresponding Lagrangians are explicitly presented. In Sec. 3, we consider F(R,T) theory,
where R and T will be generalized with respect to the usual notions of curvature scalar and torsion
scalar. Some reductions of F(R,T) gravity are presented in Sec. 4. In Sec. 5, the specific
model F(R,T) = µR + νT is analized and in Sec. 6 the exact power-law solution is found; some
cosmological implications of the model will be here discussed. The Bianchi type I version of F(R,T)
gravity is considered in Sec. 7. Sec. 8 is devoted to some generalizations of some modified gravity
theories. Final conclusions and remarks are provided in Sec. 9.
2 Preliminaries of F(R), F(G) and F(T) gravities
At the beginning, we present the basic equations of F(R), F(T) and F(G) modified gravity theories.
For simplicity we mainly work in the FRW spacetime.
2.1 F(R) gravity
The action of F(R) theory is given by
∫
S R =
d 4 xe[F(R) + L m ], (2.1)
where R is the curvature scalar. We work with the FRW metric (1.1). In this case R assumes the
form
R = R s = 6(Ḣ + 2H2 ). (2.2)
The action we rewrite as
where the Lagrangian is given by
∫
S R = dtL R , (2.3)
L R = a 3 (F − RF R ) − 6F R aȧ 2 − 6F RR Ṙa 2 ȧ − a 3 L m . (2.4)
The corresponding field equations of F(R) gravity read
2.2 F(T) gravity
6ṘHF RR − (R − 6H 2 )F R + F = ρ, (2.5)
−2Ṙ2 F RRR + [−4ṘH − 2 ¨R]F RR + [−2H 2 − 4a −1 ä + R]F R − F = p, (2.6)
˙ρ + 3H(ρ + p) = 0. (2.7)
In the modified teleparallel gravity, the gravitational action is
∫
S T = d 4 xe[F(T) + L m ], (2.8)
where e = det (e i µ) = √ −g , and for convenience we use the units 16πG = = c = 1 throughout.
The torsion scalar T is defined as
T ≡ S ρ µν T ρ µν , (2.9)
where
T ρ µν ≡ −e ρ i
(
∂µ e i ν − ∂ ν e i )
µ , (2.10)
K µν ρ ≡ − 1 2 (T µν ρ − T νµ ρ − T ρ µν ) , (2.11)
S ρ
µν
≡ 1 2
(
K
µν
ρ + δ ρT µ θν θ − δρT ν θµ )
θ . (2.12)
For a spatially flat FRW metric (1.1), as a consequence of equations (3.9) and (1.1), we have that
the torsion scalar assumes the form
T = T s = −6H 2 . (2.13)
3

The action (3.8) can be written as
where the point-like Lagrangian reads
∫
S T =
dtL T , (2.14)
The equations of F(T) gravity look like
L T = a 3 (F − F T T) − 6F T aȧ 2 − a 3 L m . (2.15)
12H 2 F T + F = ρ, (2.16)
)
48H 2 F TT Ḣ − F T
(12H 2 + 4Ḣ − F = p, (2.17)
˙ρ + 3H(ρ + p) = 0. (2.18)
2.3 F(G) gravity
The action of F(G) theory is given by
∫
S G =
d 4 xe[F(G) + L m ], (2.19)
where the Gauss – Bonnet scalar G for the FRW metric is
G = G s = 24H 2 (Ḣ + H2 ). (2.20)
3 Geometrical roots of F(R, T) gravity
We start from the M 43 - model (about our notations, see e.g. Refs. [10]-[11]). This model is one
of the representatives of F(R,T) gravity. The action of the M 43 - model reads as
∫
S 43 = d 4 x √ −g[F(R,T) + L m ],
R = R s = ɛ 1 g µν R µν , (3.1)
T = T s = ɛ 2 S ρ µν T ρ µν,
where L m is the matter Lagrangian, ɛ i = ±1 (signature), R is the curvature scalar, T is the torsion
scalar (about our notation see below). In this section we try to give one of the possible geometric
formulations of M 43 - model. Note that we have different cases related with the signature: (1)
ɛ 1 = 1,ɛ 2 = 1; (2) ɛ 1 = 1,ɛ 2 = −1; (3) ɛ 1 = −1,ɛ 2 = 1; (4) ɛ 1 = −1,ɛ 2 = −1. Also note that M 43 -
model is a particular case of M 37 - model having the form
∫
S 37 = d 4 x √ −g[F(R,T) + L m ],
where
R = u + R s = u + ɛ 1 g µν R µν , (3.2)
T = v + T s = v + ɛ 2 S ρ µν T ρ µν,
are the standard forms of the curvature and torsion scalars.
3.1 General case
R s = ɛ 1 g µν R µν , T s = ɛ 2 S ρ µν T ρ µν (3.3)
To understand the geometry of the M 43 - model, we consider some spacetime with the curvature
and torsion so that its connection G λ µν is a sum of the curvature and torsion parts. In this paper,
the Greek alphabets (µ, ν, ρ, ... = 0,1,2,3) are related to spacetime, and the Latin alphabets
4

(i,j,k,... = 0,1,2,3) denote indices, which are raised and lowered with the Minkowski metric η ij
= diag (−1,+1,+1,+1). For our spacetime the connection G λ µν has the form
G λ µν = e i λ ∂ µ e i ν + e j λ e i νω j iµ = Γ λ µν + K λ µν. (3.4)
Here Γ j iµ is the Levi-Civita connection and Kj iµ is the contorsion. Let the metric has the form
ds 2 = g ij dx i dx j . (3.5)
Then the orthonormal tetrad components e i (x µ ) are related to the metric through
so that the orthonormal condition reads as
Here η ij = diag(−1,1,1,1), where we used the relation
The quantities Γ j iµ and Kj iµ are defined as
g µν = η ij e i µe j ν, (3.6)
η ij = g µν e µ i eν j . (3.7)
e i µe µ j = δi j. (3.8)
Γ l jk = 1 2 glr {∂ k g rj + ∂ j g rk − ∂ r g jk } (3.9)
and
K λ µν = − 1 2
(
T
λ
µν + T µν λ + T νµ
λ ) , (3.10)
respectively. Here the components of the torsion tensor are given by
The curvature R ρ σµν is defined as
T λ µν = e i λ T i µν = G λ µν − G λ νµ, (3.11)
T i µν = ∂ µ e i ν − ∂ ν e i µ + G i jµe j ν − G i jνe j µ. (3.12)
R ρ σµν = e i ρ e j σR i jµν = ∂ µ G ρ σν − ∂ ν G ρ σµ + G ρ λµG λ σν − G ρ λνG λ σµ
= ¯R ρ σµν + ∂ µ K ρ σν − ∂ ν K ρ σµ + K ρ λµK λ σν − K ρ λνK λ σµ
+Γ ρ λµ Kλ σν − Γ ρ λν Kλ σµ + Γ λ σνK ρ λµ − Γ λ σµK ρ λν, (3.13)
where the Riemann curvature of the Levi-Civita connection is defined in the standard way
¯R ρ σµν = ∂ µ Γ ρ σν − ∂ ν Γ ρ σµ + Γ ρ λµ Γλ σν − Γ ρ λν Γλ σµ. (3.14)
Now we introduce two important quantities namely the curvature (R) and torsion (T) scalars as
R = g ij R ij , (3.15)
T = S ρ µν Tµν, ρ (3.16)
where
S ρ µν = 1 2
(
Kρ µν + δ µ ρT θ θν − δ ν ρT θ
θµ ) . (3.17)
Then the M 43 - model we write in the form (3.1). To conclude this subsection, we note that in
GR, it is postulated that T λ µν = 0 and such 4-dimensional spacetime manifolds with metric and
without torsion are labelled as V 4 . At the same time, it is a general convention to call U 4 , the
manifolds endowed with metric and torsion.
5

3.2 FRW case
From here we work with the spatially flat FRW metric
ds 2 = −dt 2 + a 2 (t)(dx 2 + dy 2 + dz 2 ), (3.18)
where a(t) is the scale factor. In this case, the non-vanishing components of the Levi-Civita
connection are
Γ 0 00 = Γ 0 0i = Γ 0 i0 = Γ i 00 = Γ i jk = 0,
Γ 0 ij = a 2 Hδ ij , (3.19)
Γ i jo = Γ i 0j = Hδ i j,
where H = (lna) t and i,j,k,... = 1,2,3. Now let us calculate the components of torsion tensor.
Its non-vanishing components are given by:
T 110 = T 220 = T 330 = a 2 h,
T 123 = T 231 = T 312 = 2a 3 f, (3.20)
where h and f are some real functions (see e.g. Refs. [12]). Note that the indices of the torsion
tensor are raised and lowered with respect to the metric, that is
Now we can find the contortion components. We get
T ijk = g kl T ij l . (3.21)
K 1 10 = K 2 20 = K 3 30 = 0,
K 1 01 = K 2 02 = K 3 03 = h,
K 0 11 = K 0 22 = K 0 22 = a 2 h, (3.22)
K 1 23
K 1 32
= K 2 31 = K 3 12 = −af,
= K 2 13 = K 3 21 = af.
The non-vanishing components of the curvature R ρ σµν are given by
R 0 101 = R 0 202 = R 0 303 = a 2 (Ḣ + H2 + Hh + ḣ),
R 0 123 = −R 0 213 = R 0 312 = 2a 3 f(H + h),
R 1 203 = −R 1 302 = R 2 301 = −a(Hf + f), ˙
R 1 212 = R 1 313 = R 2 323 = a 2 [(H + h) 2 − f 2 ]. (3.23)
Similarly, we write the non-vanishing components of the Ricci curvature tensor R µν as
R 00 = −3Ḣ − 3ḣ − 3H2 − 3Hh,
R 11 = R 22 = R 33 = a 2 (Ḣ + ḣ + 3H2 + 5Hh + 2h 2 − f 2 ). (3.24)
At the same time, the non-vanishing components of the tensor S µν
ρ
are given by
S1 10 = 1 (
K
10
1 + δ 1
2
1Tθ
θ0
S 10
1 = S 20
2 = S 30
S 23
1 = 1 2
− δ1T 0 θ
θν ) 1 = (h + 2h) = h, (3.25)
2
3 = 2h, (3.26)
(
K
23
1 + δ 2 1 + δ 3 1
S 23
1 = S 31
2 = S 21
3 = − f 2a
)
= −
f
2a , (3.27)
(3.28)
and
T = T 1 10S 10
1 + T 2 20S 20
2 + T 3 30S 30
3 + T 23
1 S 1 23 + T 2 31S 31
2 + T 3 12S 12
3 . (3.29)
6

Now we are ready to write the explicit forms of the curvature and torsion scalars. We have
R = 6(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 (3.30)
T = 6(h 2 − f 2 ). (3.31)
So finally for the FRW metric, the M 43 - model takes the form
∫
S 43 = d 4 x √ −g[F(R,T) + L m ],
R = 6(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 , (3.32)
T = 6(h 2 − f 2 ).
It (that is the M 43 - model) is one of geometrical realizations of F(R,T) gravity in the sense that
it was derived from the purely geometrical point of view.
4 Lagrangian formulation of F(R, T) gravity
Of course, we can work with the form (3.32) of F(R,T) gravity. But a more interesting and general
form of F(R,T) gravity is the so-called M 37 - model. The action of the M 37 - gravity reads as [10]
∫
S 37 = d 4 x √ −g[F(R,T) + L m ],
where
R = u + R s = u + 6ɛ 1 (Ḣ + 2H2 ), (4.1)
T = v + T s = v + 6ɛ 2 H 2 ,
R s = 6ɛ 1 (Ḣ + 2H2 ), T s = 6ɛ 2 H 2 . (4.2)
So we can see that here instead of two functions h and f in (3.32), we introduced two new functions
u and v. For example, for (3.32) these functions have the form
u = 6(1 − ɛ 1 )(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 , (4.3)
v = 6(h 2 − f 2 − ɛ 2 H 2 ) (4.4)
that again tells us that the M 43 - model is a particular case of M 37 - model [Note that if ɛ 1 = 1 = ɛ 2
we have u = 6ḣ + 18Hh + 6h2 − 3f 2 , v = 6(h 2 − f 2 − H 2 )]. But in general we think (or assume)
that u = u(t,a,ȧ,ä, ... a,...;f i ) and v = v(t,a,ȧ,ä, ... a,...;g i ), while f i and g i are some unknown
functions related with the geometry of the spacetime. So below we will work with a more general
form of F(R,T) gravity namely the M 37 - gravity (4.1). Introducing the Lagrangian multipliers
we now can rewrite the action (4.1) as
∫
S 37 = dta
{F(R,T) 3 ȧ
− λ
[T 2 ] [ (ä )] }
− v − 6ɛ 2
a 2 − γ R − u − 6ɛ 1
a + ȧ2
a 2 + L m , (4.5)
where λ and γ are Lagrange multipliers. If we take the variations with respect to T and R of this
action, we get
λ = F T , γ = F R . (4.6)
Therefore, the action (4.5) can be rewritten as
∫
S 37 =
dta 3 {
F(R,T) − F T
[
T − v − 6ɛ 2
ȧ 2
a 2 ]
− F R
[
R − u − 6ɛ 1
(ä
Then the corresponding point-like Lagrangian reads
a + ȧ2
a 2 )]
+ L m
}
. (4.7)
L 37 = a 3 [F − (T − v)F T − (R − u)F R + L m ] − 6(ɛ 1 F R − ɛ 2 F T )aȧ 2 − 6ɛ 1 (F RR Ṙ + F RT ˙ T)a 2 ȧ. (4.8)
7

As is well known, for our dynamical system, the Euler-Lagrange equations read as
( )
d ∂L37
− ∂L 37
= 0, (4.9)
dt ∂Ṙ ∂R
( )
d ∂L37
dt ∂T
˙ − ∂L 37
= 0, (4.10)
∂T
( )
d ∂L37
− ∂L 37
= 0. (4.11)
dt ∂ȧ ∂a
Hence, using the expressions
we get
∂L 37
∂Ṙ = −6ɛ 1F RR a 2 ȧ, (4.12)
∂L 37
∂T
˙
∂L 37
∂ȧ
respectively. Here
= −6ɛ 1 F RT a 2 ȧ, (4.13)
= −12(ɛ 1 F R − ɛ 2 F T )aȧ − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ, (4.14)
a 3 F TT
(
T − v − 6ɛ 2
ȧ 2
a 2 )
(
)
a 3 F RR R − u − 6ɛ 1 (ä
a + ȧ2
a 2 )
= 0, (4.15)
= 0, (4.16)
U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF + 6ɛ 2 a 2 p = 0, (4.17)
U = A 3 F RRR + A 2 F RR + A 1 F R , (4.18)
A 3 = −6ɛ 1 Ṙ 2 a 2 , (4.19)
A 2 = −12ɛ 1 Ṙaȧ − 6ɛ 1 ¨Ra 2 + a 3 Ṙuȧ, (4.20)
A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 ˙uȧ − a 3 u a , (4.21)
B 2 = 12ɛ 2 Taȧ ˙ + a 3 Tvȧ, ˙
(4.22)
B 1 = 24ɛ 2 ȧ 2 + 12ɛ 2 aä + 3a 2 ȧvȧ + a 3 ˙vȧ − a 3 v a , (4.23)
C 2 = −12ɛ 1 a 2 ṘT, ˙
(4.24)
C 1 = −6ɛ 1 a 2 T˙
2 , (4.25)
C 0 = −12ɛ 1 Taȧ ˙ + 12ɛ 2 Ṙaȧ − 6ɛ 1 a 2 ¨T + a 3 Ṙvȧ + a 3 Tuȧ, ˙
(4.26)
M = −3a 2 . (4.27)
If F RR ≠ 0,
F TT ≠ 0, from Eqs. (4.17) and (4.17), it is easy to find that
R = u + 6ɛ 1 (Ḣ + 2H2 ), T = v + 6ɛ 2 H 2 , (4.28)
so that the relations (4.1) are recovered. Generally, these equations are the Euler constraints of
the dynamics. Here a,R,T are the generalized coordinates of the configuration space. On the
other hand, it is also well known that the total energy (Hamiltonian) corresponding to Lagrangian
L 37 is given by
H 37 = ∂L 37
∂ȧ ȧ + ∂L 37
Ṙ + ∂L 37
∂Ṙ ∂T
˙ T˙
− L 37 . (4.29)
Hence using (4.12)-(4.14) we obtain
H 37 = [−12(ɛ 1 F R − ɛ 2 F T )aȧ − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ]ȧ
−6ɛ 1 F RR a 2 ȧṘ − 6ɛ 1F RT a 2 ȧ ˙ T − [a 3 (F − TF T − RF R + vF T + uF R + L m )−
6(ɛ 1 F R − ɛ 2 F T )aȧ 2 − 6ɛ 1 (F RR Ṙ + F RT ˙ T + F Rψ ˙ψ)a 2 ȧ]. (4.30)
8

Let us rewrite this formula as
where
H 37 = D 2 F RR + D 1 F R + JF RT + E 1 F T + KF + 2a 3 ρ, (4.31)
D 2 = −6ɛ 1 Ṙa 2 ȧ, (4.32)
D 1 = 6ɛ 1 aä + a 3 uȧȧ, (4.33)
J = −6ɛ 1 a 2 ȧT, ˙
(4.34)
E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, (4.35)
K = −a 3 . (4.36)
As usual we assume that the total energy H 37 = 0 (Hamiltonian constraint). So finally we have
the following equations of the M 37 - model [10]-[11]:
D 2 F RR + D 1 F R + JF RT + E 1 F T + KF = −2a 3 ρ,
U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF = 6a 2 p, (4.37)
˙ρ + 3H(ρ + p) = 0.
It deserves to note that the M 37 - model (4.1) admits some interesting particular and physically
important cases. Some particular cases are now presented.
i) The M 44 - model. Let the function F(R,T) be independent from the torsion scalar T:
F = F(R,T) = F(R). Then the action (4.1) acquires the form
∫
S 44 = d 4 xe[F(R) + L m ], (4.38)
where
R = u + R s = u + ɛ 1 g µν R µν , (4.39)
is the curvature scalar. It is the M 44 - model. We work with the FRW metric. In this case R takes
the form
R = u + 6ɛ 1 (Ḣ + 2H2 ). (4.40)
The action can be rewritten as
where the Lagrangian is given by
∫
S 44 = dtL 44 , (4.41)
L 44 = a 3 [F − (R − u)F R + L m ] − 6ɛ 1 F R aȧ 2 − 6ɛ 1 F RR Ṙa 2 ȧ. (4.42)
The corresponding field equations of the M 44 - model read as
Here
and
D 2 F RR + D 1 F R + KF = −2a 3 ρ,
A 3 F RRR + A 2 F RR + A 1 F R + MF = 6a 2 p, (4.43)
˙ρ + 3H(ρ + p) = 0.
D 2 = −6ɛ 1 Ṙa 2 ȧ, (4.44)
D 1 = 6ɛ 1 a 2 ä + a 3 uȧȧ, (4.45)
K = −a 3 (4.46)
A 3 = −6ɛ 1 Ṙ 2 a 2 , (4.47)
A 2 = −12ɛ 1 Ṙaȧ − 6ɛ 1 ¨Ra 2 + a 3 Ṙuȧ, (4.48)
A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 ˙uȧ − a 3 u a , (4.49)
M = −3a 2 . (4.50)
9

If u = 0 then we get the following equations of the standard F(R s ) gravity (after R = R s ):
6ṘHF RR − (R − 6H 2 )F R + F = ρ, (4.51)
−2Ṙ2 F RRR + [−4ṘH − 2 ¨R]F RR + [−2H 2 − 4a −1 ä + R]F R − F = p, (4.52)
˙ρ + 3H(ρ + p) = 0. (4.53)
ii) The M 45 - model. The action of the M 45 - model looks like
∫
S 45 = d 4 xe[F(T) + L m ], (4.54)
where e = det (e i µ) = √ −g and the torsion scalar T is defined as
Here
T = v + T s = v + ɛ 2 S ρ µν T ρ µν. (4.55)
T ρ µν ≡ −e ρ i
(
∂µ e i ν − ∂ ν e i )
µ , (4.56)
K µν ρ ≡ − 1 2 (T µν ρ − T νµ ρ − T ρ µν ) , (4.57)
S ρ
µν
≡ 1 2
(
K
µν
ρ + δ ρT µ θν θ − δρT ν θµ )
θ . (4.58)
For a spatially flat FRW metric (3.18), we have the torsion scalar in the form
T = v + T s = v + 6ɛ 2 H 2 . (4.59)
The action (4.54) can be written as
where the point-like Lagrangian reads
∫
S 45 =
dtL 45 , (4.60)
L 45 = a 3 [F − (T − v)F T + L m ] + 6ɛ 2 F T aȧ 2 . (4.61)
So finally we get the following equations of the M 45 - model:
E 1 F T + KF = −2a 3 ρ,
B 2 F TT + B 1 F T + MF = 6a 2 p, (4.62)
˙ρ + 3H(ρ + p) = 0.
Here
E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, (4.63)
K = −a 3 (4.64)
and
B 2 = 12ɛ 2 ˙ Taȧ + a 3 ˙ Tvȧ, (4.65)
B 1 = 24ɛ 2 ȧ 2 + 12ɛ 2 aä + 3a 2 ȧvȧ + a 3 ˙vȧ − a 3 v a , (4.66)
M = −3a 2 . (4.67)
If we put v = 0 then the M 45 - model reduces to the usual F(T s ) gravity, where T s = 6ɛ 2 H 2 . As
is well-known the equations of F(T s ) gravity are given by
12H 2 F T + F = ρ, (4.68)
)
48H 2 F TT Ḣ − F T
(12H 2 + 4Ḣ − F = p, (4.69)
˙ρ + 3H(ρ + p) = 0, (4.70)
where we must put T = T s . Finally we note that it is well-known that the standard F(T s ) gravity
is not local Lorentz invariant [33]. In this context, we have a very meager hope that the M 45 -
model (4.54) is free from such problems.
10

5 Cosmological solutions
In this section we investigate cosmological consequences of the F(R,T) gravity. As example, we
want to find some exact cosmological solutions of the M 37 - gravity model. Since its equations are
very complicated we here consider the simplest case when
F(R,T) = µR + νT, (5.1)
where µ and ν are some constants. Then equations (4.37) take the form
where
We can rewrite this system as
µD 1 + νE 1 + K(νT + µR) = −2a 3 ρ,
µA 1 + νB 1 + M(νT + µR) = 6a 2 p, (5.2)
˙ρ + 3H(ρ + p) = 0,
D 1 = 6ɛ 1 a 2 ä + a 3 uȧȧ, (5.3)
E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, (5.4)
K = −a 3 , (5.5)
A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 ˙uȧ − a 3 u a , (5.6)
B 1 = 24ɛ 2 ȧ 2 + 12ɛ 2 aä + 3a 2 ȧvȧ + a 3 ˙vȧ − a 3 v a , (5.7)
M = −3a 2 . (5.8)
3σH 2 − 0.5(ȧzȧ − z) = ρ,
−σ(2Ḣ + 3H2 ) + 0.5(ȧzȧ − z) + 1 6 a(ż ȧ − z a ) = p, (5.9)
˙ρ + 3H(ρ + p) = 0,
where z = µu + νv, σ = µɛ 1 − νɛ 2 . This system contents two independent equations for five
unknown functions (a,ρ,p,u,v). But in fact it contain 4 unknown functions (H,ρ,p,z). The
corresponding EoS parameter reads as
ω = p ρ = −1 + −2σḢ + 1 6 a(ż ȧ − z a )
3σH 2 − 0.5(ȧzȧ − z) . (5.10)
Let us find some simplest cosmological solutions of the system (5.9).
5.1 Example 1
We start from the case σ = 0. In this case the system (5.9) takes the form
At the same time the EoS parameter becomes
Now we assume that
where κ and l are some real constants. Then
−0.5(ȧzȧ − z) = ρ,
0.5(ȧzȧ − z) + 1 6 a(ż ȧ − z a ) = p, (5.11)
˙ρ + 3H(ρ + p) = 0.
ω = p ρ = −1 − a(ż ȧ − z a )
3(ȧzȧ − z) . (5.12)
z = κa l , (5.13)
ω = −1 − l 3 . (5.14)
This result tells us that in this case our model can describes the accelerated expansion of the
Universe for some values of l.
11

5.2 Example 2
Now consider the de Sitter case that is H = H 0 = const so that a = a 0 e H0t . Then the system
(5.9) reads as
The EoS parameter takes the form
3σH 2 0 − 0.5(ȧzȧ − z) = ρ,
−3σH 2 0 + 0.5(ȧzȧ − z) + 1 6 a(ż ȧ − z a ) = p, (5.15)
If z has the form (5.13) that is z = κe H0lt then we have
˙ρ + 3H(ρ + p) = 0.
ω = p ρ = −1 + a(żȧ − z a )
18σH 2 0 − 3(ȧz ȧ − z) . (5.16)
ω = p ρ = −1 −
If H 0 l > 0 then as t → ∞ we get again
lκ
18σH 2 0 e−H0lt + 3κ . (5.17)
ω = −1 − l 3 , (5.18)
which is same as equation (5.14). This last equation tells us that our model can describes the
accelerated expansion of the Universe e.g. for l ≥ 0. Also it corresponds to the phantom case if
l > 0. Finally we present other forms of the generalized Friedmann equations in the system (5.9).
Let us rewrite these equations in the standard form as
Here
3H 2 = ρ + ρ z , (5.19)
−(2Ḣ + 3H2 ) = p + p z . (5.20)
ρ z = 0.5σ −1 (ȧzȧ − z), (5.21)
p z = −0.5σ −1 (ȧzȧ − z) − 1
6σ a(ż ȧ − z a ) (5.22)
are the z or u − v contributions to the energy density and pressure, respectively.
6 Conclusion
In [30], Buchdahl proposed to replace the Einstein-Hilbert scalar Lagrangian R with a function of
the scalar curvature. The resulting theory is nowadays known as F(R) gravity. Almost 40 years
later, Bengochea and Ferraro proposed to replace the TEGR that is the torsion scalar Lagrangian
T with a function F(T) of the torsion scalar, and studied its cosmological consequences [31]. This
type of modified gravity is nowadays called as F(T) gravity theory. These two gravity theories
[that is F(R) and F(T)] are, in some sense, alternative ways to modify GR. From these results
arises the natural question: how we can construct some modified gravity theory which unifies F(R)
and F(T) theories Examples of such unified curvature-torsion theories were proposed in [10]-[11].
Such type of modified gravity theory is called the F(R,T) gravity. In this F(R,T) gravity, the
curvature scalar R and the torsion scalar T play the same role and are dynamical quantities. In
this paper, we have shown that the F(R,T) gravity can be derived from the geometrical point of
view. In particular, we have proposed a new method to construct particular models of F(R,T)
gravity. As an example we have considered the M 43 - model, deriving its action in terms of the
curvature and torsion scalars. Then in detail we have studied the M 37 - model and presented its
action, Lagrangian and equations of motion for the FRW metric case. Finally we have shown that
the last model can describes the accelerated expansion of the Universe.
12

Concluding, we would like to note that in the paper, we present a special class of extended
gravity models depending on arbitrary function F(R,T), where R is the Ricci scalar and T the
scalar torsion. While in the traditional Eistein-Cartan theory, the role of the torsion depends on
the non trivial source associated with spin matter density, in our F(R,T) gravity models, the
torsion can propagate without the presence of spin matter density. In fact this is a crucial point,
otherwise the additional scalar torsion degree of freedom are not different from the additional
metric gravitational degree of freedom present in extended F(R) models. Finally we would like to
note that all results of this paper are new and different than results of our previous papers [10]-[11]
on the subject.
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